Complexity of the variety of orthogonal models for irreps of Weyl groups This table reports the number of equations and variables that define the variety of solutions for the matrix representing s[n] in a hereditary orthogonal model for each irrep of the Weyl groups of E6, E7, and E8. These counts do not include equations that are required to eliminate solutions corresponding to clones, nor the vanishing conditions required for the representations that are not totally free (marked by *'s). Column 2 is the dimension of the representation. Column 3 is the number of equations. column 4 is the number of variables. E6 1 1 2 1 2 1 2 1 3 6 12 4 4 6 12 4 5 10 10 2 6 15 25 5 7 15 25 5 8 15 25 5 9 15 25 5 10 20 58 13 11 20 58 13 12 20 31 5 13 24 41 6 14 24 41 6 15 30 79 13 16 30 79 13 17 60 217 28 18 60 217 28 19 60 199 25 20 64 274 37 21 64 274 37 22 80 330 36 23 81 355 40 24 81 355 40 25 90 417 45 E7 1 1 2 1 2 1 2 1 3 7 11 4 4 7 11 4 5 15 7 2 6 15 7 2 7 21 23 6 8 21 23 6 9 21 18 5 10 21 18 5 11 27 43 11 12 27 43 11 13 35 33 7 14 35 33 7 15 35 19 5 16 35 19 5 17 56 77 14 18 56 77 14 19 70 44 7 20 70 44 7 21 84 51 8 22 84 51 8 23 105 159 24 24 105 159 24 25 105 68 9 26 105 68 9 27 105 107 15 28 105 107 15 29 120 195 27 30 120 195 27 31 168 240 28 32 168 240 28 33 189 272 30 34 189 272 30 35 189 338 39 36 189 338 39 37 189 229 26 38 189 229 26 39 210 263 28 40 210 263 28 41 210 374 41 42 210 374 41 43 216 285 30 44 216 285 30 45 280 477 45 46 280 477 45 47 280 482 45 48 280 482 45 49 315 585 51 50 315 585 51 51 336 586 50 52 336 586 50 53 378 742 61 54 378 742 61 55 405 884 71 56 405 884 71 57 420 874 68 58 420 874 68 59 512 1260 93 60 512 1260 93 E8 1 1 2 1 2 1 2 1 3 8 11 4 4 8 11 4 5 28 18 5 6 28 18 5 7 35 39 10 8 35 39 10 9 50 16 4 10 50 16 4 11 56 19 5 12 56 19 5 13 70 19 5 14 84 69 14 15 84 69 14 16 112 123 23 17 112 123 23 18 160 121 20 19 160 121 20 20 168 28 5 21 175 38 6 22 175 38 6 23 210 162 23 24 210 162 23 25 300 139 16 26 300 139 16 27 350 149 21 28 350 149 21 29 400 226 27 30 400 226 27 31 420 103 11 32 448 151 21 33 448 192 20 34 448 192 20 35 525 204 20 36 525 204 20 37 560 622 66 38 560 622 66 39 567 622 67 40 567 622 67 41 700 609 57 42 700 609 57 43 700 250 25 44 700 250 25 45 840 359 29 46 840 359 29 47 840 323 27 48 840 323 27 49 972 519 42 50 972 519 42 51 1008 962 80 52 1008 962 80 53 1050 714 59 54 1050 714 59 55 1134 436 31 56 1296 1015 79 57 1296 1015 79 58 1344 1500 113 59 1344 1500 113 60 1344 693 51 61 1400 1005 74 62 1400 1005 74 63 1400 1165 80 64 1400 1165 80 65 1400 1637 121 66 1400 1637 121 67 1400 641 37 68 1575 1553 107 69 1575 1553 107 70 1680 1109 80 71 2016 1447 88 72 2100 1789 110 73 2100 1789 110 74 2100 1433 84 75 2240 2234 132 76 2240 2234 132 77 2268 2491 145 78 2268 2491 145 79 2400 1997 114 80 2400 1997 114 81 2688 2185 111 82 2800 2829 148 83 2800 2829 148 84 2835 2763 147 85 2835 2763 147 86 3150 3068 152 87 3200 3174 157 88 3200 3174 157 89 *3240 5299 298 90 *3240 5299 298 91 3360 3907 191 92 3360 3907 191 93 4096 6311 305 94 4096 6311 305 95 4096 6311 305 96 4096 6311 305 97 4200 6048 282 98 4200 6048 282 99 4200 5414 249 100 4200 5414 249 101 4200 5365 252 102 4480 5816 254 103 *4536 6870 327 104 *4536 6870 327 105 4536 5899 253 106 5600 8879 368 107 *5600 9350 402 108 *5600 9350 402 109 5670 9131 380 110 *6075 11253 477 111 *6075 11253 477 112 *7168 14597 593